10 1 Composition of Functions

追加: 2026-05-11

[00:00:05]okay while we're talking about different types of function notation we are going to jump to 10.1 and talk about composition of functions more function notation okay so let's uh look at an example of where g of x is given as two x squared over x minus one and f of x is given as x squared minus one and you will be asked to compose these functions so something that they may ask you to do is to find g of f of x g of f of x or you may see it written as g of f of x those two notations mean the same thing and we're used to substituting in numbers like g of 4 and find a result but in this case we're substituting in an expression or another function so what is f of x well f of x is x squared minus one so in reality it's asking us to find g of x squared minus one well how do we do that well again this is our input this is our input this is our replacement for every x in g it's our replacement so let's go ahead and replace that replace this x and replace this x with our input so it's going to be 2 times x squared minus 1 squared all over x squared minus one again replacement with our input minus one okay so that's your answer now let's just simplify it to square a binomial you have to foil the two binomials together or you can remember how to square binomial using the binomial square theorem or pascal's triangle which is what i would do it's going to save you time if you know this in the future so to square a binomial it's the first term squared the first term times the second term times two and then the second one squared and on the bottom we have x squared minus one minus one so your final answer is two x to the fourth minus four x squared plus two all over x squared minus two now let's take a look at something a little bit different let's take a look at an example where they say given that f of x equals x squared minus one this should look familiar to you find f of x plus h minus f of x all over h remember that i'm going to practice that some more because that comes up in calculus all the time okay so this is f of x plus h and this is f of x so the question is how do you find f of x plus h well again this is our input this is our input into f it's our replacement into f for every x so let's go ahead and do that we're replacing our x with our input x plus h okay so that is f of x plus h that's just this first piece here then we need to subtract the f of x as i've said before let's simplify this first again squaring a binomial the first term squared the first times the second times two and then the second one squared so we've done this before just doing more practice so let's now find the original f of x plus h minus f of x all over h so this is our f of x plus h i figured out that piece first and simplified it x squared plus two x h plus h squared minus one there's our f of x plus h minus our f of x well what's our f of x our f of x is given you have to subtract the quantity otherwise you're going to get the wrong answer if you don't use parentheses and then all divided by h okay so that's going to give you x squared minus x squared which is zero and then you're going to have 2xh minus no 2xhs and then plus h squared minus nothing and then a negative one minus a negative one is the same thing as a negative one plus one which is zero and that's all divided by h now if you recall each term has an h in it so you can cancel out an h out of each or to show your work properly factor out an h out of each term and now you can cancel the h's and your final answer is 2x plus h so that's practicing the difference quotient okay understanding function notation that's what we've been doing last couple of sections okay now another way that they need you to understand function notation and then three is what if i give you f of x as a composite function i'm giving it to you composed already and i am going to tell you that this is the composition of u of v of x so i'm telling you that you inputted v of x into u you replaced v of x for every x that was in u so can you think backwards and try to figure out what v of x is and what u of x would have to be if this is the composition what did you plug in what did you substitute in your substitution looking at this i substituted in this is my inside i substitute in x cubed plus one and what was the function that i substituted or replaced all of this with each x what would have been the original equation with the x in it well if i substituted all of this in for x and i replaced an x then my original function must have been x squared and that's called decomposing that's the opposite of composition decomposing so i think we need to try a couple more example number four let's say h of x is equal to oh let's go with 11x squared plus 12x the quantity squared and let me tell you that that composition came about by composing f of g of x okay f of g of x is 11 x squared plus 12 x the quantity squared what is your input so i would say 11x squared plus 12x was my input i replaced that with all x of what function well again my outer function i plugged in all of this into f so f of x must have been x squared okay so that's just like that one let's try ones that are a little bit different let's say h of x was your composition and your end result was x squared plus two one over x squared plus two okay and that was the composition of f of g of x so this is f of g of x our composition resulted in this see if you can figure that one out you can stop the video and give it a try okay the one thing you cannot say is i plugged x into one over x squared plus two okay that that would be useless i could say that for every single problem so you can't you can't use that so what could i have what could have been my substitution well my substitution my replacement could have been x squared plus two that means that i replaced x squared plus two for every x in my f function so if i replaced my x with this what was the original function what was f of x this is your substitution into the original this is the composition i substituted x squared plus two into one over x okay decomposing all right one more let's say that our composition our end result was x minus 2 the quantity squared plus 1 all over 5 minus x minus 2 the quantity squared okay and that was a result of f of g of x or again notation f of g of x that's another notation for composition when you see it like this it's you're plugging g of x into f you're plugging g of x into f so looking at this as our end result what do you think g of x is what do you think you substituted into the original well really there's sometimes there's two possibilities so the first one i'm gonna go with is i substituted in all of this i substituted in x minus two squared into x plus one over five minus x if i substitute all of this in for x i get this if i substitute all of this in for x i get this so that's one possibility can you see another possibility how about if my substitution was just x minus two how about my replacement was just x minus two go ahead and figure out what equation you would have substituted what function would you have substituted x minus 2 into to get that this is your replacement for x so thinking backwards if this was my replacement for x then it should be x squared plus one over five minus x squared so there's two options for this one sometimes there are multiple decompositions depends on what you see again you cannot say i plugged in x that is not an answer that you can give for the decompositions you